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This thread has devolved quickly into the Frequentist vs Bayesian debate, and that is not easily resolvable. The math in both approaches is solid, so it always comes down to philosophical preferences. The frequentist interpretation of probability as the limit of an event's relative frequency is justified by the strong law of large numbers; regardless of your preferred interpretation of probability, an event's relative frequency will converge to its probability with probability 1.

Frequentist confidence intervals are indeed trickier to interpret than Bayesian credible intervals. By treating an unknown quantity as a random variable, Bayesians can assert that one interval contains that quantity with some probability. Frequentists refuse to treat some quantities as random variables, and any equations containing only constants can only be true or false. So when estimating an unknown constant, frequentists must bound them with a RANDOM interval to involve probability at all. Rather than one interval containing a random variable with some probability, a frequentist method generates many different possible intervals, some of which contains the unknown constant. If the coverage probability is reasonably high, it's a reasonable leap of faith to assert that a particular confidence interval contains the unknown constant (note, not "with some probability").

A Bayesian would balk at such a leap of faith as much as a Frequentist balks at treating any unknown quantity as a random variable. The frequentist Neyman construction method in fact exposed an embarrassing issue with such leaps of faith. Without actively preventing it (see Feldman and Cousins, 1997 for one approach), rare outcomes of a random sample may generate EMPTY confidence intervals for a distribution parameter. Such a leap of faith would be very unreasonable! I've seen a few Bayesians using that example to mock frequentist methods, while frequentists typically respond with "well I still get a correct interval most of the time, and without making false assumptions." I'll point out that the Bayesian/frequentist impasse is not important to most who apply their methods. Even people who are committed to coverage probability will often use Bayesian methods if the methods are shown to have good coverage probability in simulations.

This thread has devolved quickly into the Frequentist vs Bayesian debate, and that is not easily resolvable. The math in both approaches is solid, so it always comes down to philosophical preferences. The frequentist interpretation of probability as the limit of an event's relative frequency is justified by the strong law of large numbers; regardless of your preferred interpretation of probability, an event's relative frequency will converge to its probability with probability 1.

Frequentist confidence intervals are indeed trickier to interpret than Bayesian credible intervals. By treating an unknown quantity as a random variable, Bayesians can assert that one interval contains that quantity with some probability. Frequentists refuse to treat some quantities as random variables, and any equations containing only constants can only be true or false. So when estimating an unknown constant, frequentists must bound them with a RANDOM interval to involve probability at all. Rather than one interval containing a random variable with some probability, a frequentist method generates many different possible intervals, some of which contains the unknown constant. If the coverage probability is reasonably high, it's a reasonable leap of faith to assert that a particular interval contains the unknown constant (note, not "with some probability").

A Bayesian would balk at such a leap of faith as much as a Frequentist balks at treating any unknown quantity as a random variable. The frequentist Neyman construction method in fact exposed an embarrassing issue with such leaps of faith. Without actively preventing it (see Feldman and Cousins, 1997 for one approach), rare outcomes may generate EMPTY confidence intervals for a distribution parameter. Such a leap of faith would be very unreasonable! I've seen a few Bayesians using that example to mock frequentist methods, while frequentists typically respond with "well I still get a correct interval most of the time, and without making false assumptions." I'll point out that the Bayesian/frequentist impasse is not important to most who apply their methods. Even people who are committed to coverage probability will often use Bayesian methods if the methods are shown to have good coverage probability in simulations.

This thread has devolved quickly into the Frequentist vs Bayesian debate, and that is not easily resolvable. The math in both approaches is solid, so it always comes down to philosophical preferences. The frequentist interpretation of probability as the limit of an event's relative frequency is justified by the strong law of large numbers; regardless of your preferred interpretation of probability, an event's relative frequency will converge to its probability with probability 1.

Frequentist confidence intervals are indeed trickier to interpret than Bayesian credible intervals. By treating an unknown quantity as a random variable, they can assert that one interval contains that quantity with some probability. Frequentists refuse to treat some quantities as random variables, and any equations containing only constants can only be true or false. So when estimating an unknown constant, frequentists must bound them with a RANDOM interval to involve probability at all. Rather than one interval containing a random variable with some probability, a frequentist method generates many different possible intervals, some of which contains the unknown constant. If the coverage probability is reasonably high, it's a reasonable leap of faith to assert that a particular confidence interval contains the unknown constant (note, not "with some probability").

A Bayesian would balk at such a leap of faith as much as a Frequentist balks at treating any unknown quantity as a random variable. The frequentist Neyman construction method in fact exposed an embarrassing issue with such leaps of faith. Without actively preventing it (see Feldman and Cousins, 1997 for one approach), rare outcomes of a random sample may generate EMPTY confidence intervals for a distribution parameter. Such a leap of faith would be very unreasonable! I've seen a few pro-Bayesian arguments using that example to mock frequentist methods, while frequentists typically respond with "well I still get a correct interval most of the time, and without making false assumptions." I'll point out that the Bayesian/frequentist divide is not important to most who apply their methods. Even people who are committed to coverage probability will often use Bayesian methods if the methods are shown to have good coverage probability in simulations.

This thread has devolved quickly into the Frequentist vs Bayesian debate, and that is not easily resolvable. The math in both approaches is solid, so it always comes down to philosophical preferences. The frequentist interpretation of probability as the limit of an event's relative frequency is justified by the strong law of large numbers; regardless of your preferred interpretation of probability, an event's relative frequency will converge to its probability with probability 1.

Frequentist confidence intervals are indeed trickier to interpret than Bayesian credible intervals. By treating an unknown quantity as a random variable, Bayesians can assert that one interval contains that quantity with some probability. Frequentists refuse to treat some quantities as random variables, and any equations containing only constants can only be true or false. So when estimating an unknown constant, frequentists must bound them with a RANDOM interval to involve probability at all. Rather than one interval containing a random variable with some probability, a frequentist method generates many different possible intervals, some of which contains the unknown constant. If the coverage probability is reasonably high, it's a reasonable leap of faith to assert that a particular confidence interval contains the unknown constant (note, not "with some probability").

A Bayesian would balk at such a leap of faith as much as a Frequentist balks at treating any unknown quantity as a random variable. The frequentist Neyman construction method in fact exposed an embarrassing issue with such leaps of faith. Without actively preventing it (see Feldman and Cousins, 1997 for one approach), rare outcomes of a random sample may generate EMPTY confidence intervals for a distribution parameter. Such a leap of faith would be very unreasonable! I've seen a few Bayesians using that example to mock frequentist methods, while frequentists typically respond with "well I still get a correct interval most of the time, and without making false assumptions." I'll point out that the Bayesian/frequentist impasse is not important to most who apply their methods. Even people who are committed to coverage probability will often use Bayesian methods if the methods are shown to have good coverage probability in simulations.

This thread has devolved quickly into the Frequentist vs Bayesian debate, and that is not easily resolvable. The math in both approaches is solid, so it always comes down to philosophical preferences. The frequentist interpretation of probability as the limit of an event's relative frequency is justified by the strong law of large numbers; regardless of your preferred interpretation of probability, an event's relative frequency will converge to its probability with probability 1.

Frequentist confidence intervals are indeed trickier to interpret than Bayesian credible intervals. By treating an unknown quantity as a random variable, they can assert that one interval contains that quantity with some probability. Frequentists refuse to treat some quantities as random variables, and any equations containing only constants can only be true or false. So when estimating an unknown constant, frequentists must bound them with a RANDOM interval to involve probability at all. Rather than one interval containing a random variable with some probability, a frequentist method generates many different possible intervals, some of which contains the unknown constant. If the coverage probability is reasonably high, it's a reasonable leap of faith to assert that a particular confidence interval contains the unknown constant (note, not "with some probability").

A Bayesian would balk at such a leap of faith as much as a Frequentist balks at treating any unknown quantity as a random variable. The frequentist Neyman construction method in fact exposed an embarrassing issue with such leaps of faith. Without actively preventing it (see Feldman and Cousins, 1997 for one approach), rare outcomes of a random sample may generate EMPTY confidence intervals for a distribution parameter, even if the coverage probability is high. Such a leap of faith would be very unreasonable!

This thread has devolved quickly into the Frequentist vs Bayesian debate, and that is not easily resolvable. The math in both approaches is solid, so it always comes down to philosophical preferences. The frequentist interpretation of probability as the limit of an event's relative frequency is justified by the strong law of large numbers; regardless of your preferred interpretation of probability, an event's relative frequency will converge to its probability with probability 1.

Frequentist confidence intervals are indeed trickier to interpret than Bayesian credible intervals. By treating an unknown quantity as a random variable, they can assert that one interval contains that quantity with some probability. Frequentists refuse to treat some quantities as random variables, and any equations containing only constants can only be true or false. So when estimating an unknown constant, frequentists must bound them with a RANDOM interval to involve probability at all. Rather than one interval containing a random variable with some probability, a frequentist method generates many different possible intervals, some of which contains the unknown constant. If the coverage probability is reasonably high, it's a reasonable leap of faith to assert that a particular confidence interval contains the unknown constant (note, not "with some probability").

A Bayesian would balk at such a leap of faith as much as a Frequentist balks at treating any unknown quantity as a random variable. The frequentist Neyman construction method in fact exposed an embarrassing issue with such leaps of faith. Without actively preventing it (see Feldman and Cousins, 1997 for one approach), rare outcomes of a random sample may generate EMPTY confidence intervals for a distribution parameter. Such a leap of faith would be very unreasonable! I've seen a few pro-Bayesian arguments using that example to mock frequentist methods, while frequentists typically respond with "well I still get a correct interval most of the time, and without making false assumptions." I'll point out that the Bayesian/frequentist divide is not important to most who apply their methods. Even people who are committed to coverage probability will often use Bayesian methods if the methods are shown to have good coverage probability in simulations.